A quasi-polynomial bound for the minimal excluded minors for a surface

As part of their graph minor project, Robertson and Seymour showed in 1990 that the class of graphs that can be embedded in a given surface can be characterized by a finite set of minimal excluded minors. However, their proof, because existential, does not provide any information on these excluded minors. Seymour proved in 1993 the first and, until now, only known upper bound on the order of the minimal excluded minors for a given surface. This bound is double exponential in the Euler genus $g$ of the surface and, therefore, very far from the $\Omega(g)$ lower bound on the maximal order of minimal excluded minors for a surface and most likely far from the best possible bound. More than thirty years later, this paper finally makes progress in lowering this bound to a quasi-polynomial in the Euler genus of the surface. The main catalyzer to reach a quasi-polynomial bound is a breakthrough on the characteristic size of a forbidden structure for a minimal excluded minor $G$ for a surface of Euler genus $g$: although it is not hard to show that $G$ does not contain $O(g)$ disjoint cycles that are contractible and nested in some embedding of $G$ as demonstrated by Seymour, this bound can be lowered to $O(\log g)$ which is essential to obtain the quasi-polynomial bound in this paper. Moreover, we find an upper bound on the maximum degree of $G$ and the maximum size of a face in an embedding of $G$ in a surface of Euler genus $g + 1$ or $g + 2$, which is, to our understanding, the first such bound. Finally, we develop a new method to bound the height of the tree in a tree decomposition of $G$. As subsidiary results, we also improve the current bound on the treewidth of a minimal excluded minor $G$ for a surface by improving the first and, until now, only known bound provided by Seymour. Moreover, we show a better upper bound on the order of a grid minor in $G$, improving the result by Thomassen from 1997.